You have found the following ages (in years) of 5 sloths. The sloths are randomly selected from the 47 sloths at your local zoo: $ 8,\enspace 11,\enspace 13,\enspace 11,\enspace 11$ Based on your sample, what is the average age of the sloths? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 47 sloths, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $5$ samples and divide by $5$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{5}} x_i}{{5$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{7.84} + {0.04} + {4.84} + {0.04} + {0.04}} {{5 - 1}} $ $ {s^2} = \dfrac{{12.8}}{{4}} = {3.2\text{ years}^2} $ We can estimate that the average sloth at the zoo is 10.8 years old. There is a variance of 3.2 years $^2$.